Wiener process
The Wiener process
- also called Brownian motion, is a kind of Markov stochastic process.
- Stochastic process:
- whose value changes over time in an uncertain way
- we only know the distribution of the possible values of the process at any time point. - In contrast to the stochastic process, a deterministic process is with an exact value at any time point.
- Markov process: the likelihood of the state at any future time point depends only on
its present state but not on any past states
The properties of the Wiener process ${Z(t)}$ for $t ≥ 0$:
- (Normal increments) $Z(t) − Z(s) ∼ N(0,t − s)$.
- (Independence of increments) $Z(t) − Z(s)$ and $Z(u)$ are independent, for $u ≤ s < t$.
- (Continuity of the path) $Z(t)$ is a continuous function of t.
- $W(0) = 0$.
- $E(W(t)) = 0, E(W^2(t)) = t$ for each time $t ≥ 0$.
Stochastic differential equation (SDE)
- is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process.
- SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process.
- A typical SDE: $$dX = a(t, X) dt + b(t, X) dW_t$$
- The solution to SDE could be: Euler-Maruyama Method, Milstein Method, Runge-Kutta Method and Taylor Method
Reference:
http://math.gmu.edu/~tsauer/pre/sde.pdf
http://ft-sipil.unila.ac.id/dbooks/AN%20INTRODUCTION%20TO%20STOCHASTIC%20DIFFERENTIAL%20EQUATIONS%20VERSION%201.2.pdf